As systems and their models become more complex and costly to run, the use of tolerance limit uncertainly characterization is gaining popularity.

For example in very complex models containing several uncertain parameters (each represented by a probability distribution function), classical Bayes’ and bootstrap Monte Carlo simulation may become impractical.

Often in complex computer-based models of (5.1) in which calculation of values require significant amount of time and effort, the traditional Monte Carlo simulation is not possible.

Wilks Tolerance limit is used in these cases

A tolerance interval is a random interval (L, U) that contains with probability (or confidence) b at least a fraction g of the population under study.

The probability and fraction b and g are analyst’s selected criteria depending on the confidence desired.

The pioneering work in this area is attributed to Wilks [1-2] and later to Wald [3-4].

Wilks Tolerance limit is an efficient and simple sampling method to reduce sample size from few thousands to around 100 or so.

The number of sample size does not depend on the number of uncertain parameters in the model.

There are two kinds of tolerance limits:

**Non-parametric tolerance limits**: Nothing is known about distribution of the random variable except that it is continuous

**Parametric tolerance limits**: The distribution function representing the random variable of interest is known and only some distribution parameters involved are unknown.

The problem in both cases is to calculate a tolerance range (L, U) for a random variable X represented by the observed sample, x1, ¼, xm, and the corresponding size of the sample.

where, f(x) is the probability density function of the random variable X.

Let us consider a complex system represented by a model (e.g., a risk model).

Such a model may describe relationship between the output variables (e.g., probability of failure or performance value of a system) as a function of some input (random) variables (e.g., geometry, material properties, etc.).

Assume several parametric variables involve in the model.

Further assume that the observed randomness of the output variables is the result of the randomness of input variables.

If we take N samples of each input variable, then we obtain a sample of N output values {y1, ¼, yN} for y = f(x).

In using (1) for this problem, note that probability B bears the name confidence level.

To be on the conservative side, one should also specify probability content Y in addition to the confidence level B as large as possible.

It should be emphasized that Y is not a probability, although it is a non-negative real number of less than one [5].

Having fixed B and Y; it becomes possible to determine the number of runs (samples of output) N required to remain consistent with the selected B and Y values.

Let y1,¼,yN be N independent output values of y. Suppose that nothing is known about the pdf g(y) except that it is continuous.

Arrange the values of y1,¼, yN in an increasing order and denote them by y(k), hence

and by definition y(0) = – ∞; while y(N+1) = +∞, it can be shown that for confidence level B [5] is obtained from

From equation (3) sample sizes N can be estimated. For application of this approach consider two cases of the tolerance limits:

one-sided and two-sided follow:

**One-sided Tolerance Limits**: This is the more common case, for example when measuring a model output value such a temperature or sheer stress at a point on the surface of a structure.

We are interested in assuring that a small sample, of for example estimated temperatures, obtained from the model, and the corresponding upper sample tolerance limit TU according to (3), contains

with probability β (say 95%) at least the fraction γ of the temperatures in a fictitious sample containing infinite estimates

of such temperatures.

Table I shows values for sample size N based on values of β and γ. For example, if β = 0:95; γ = 0:90; then N = 45 samples taken from the model (e.g., by standard Monte Carlo sampling) assures that the highest temperature TH in this sample represent the 95% upper confidence limit below which 90% of the all possible temperatures lie.

**Two-Sided Tolerance Limits**: We now consider the two-sided case, which is less common [6].

Table II shows the Wilks’ sample size. With B and γ both equal to 95%, we will get N = 93 samples.

For example, in the 93 samples taken from the model (e.g., by standard Monte Carlo sampling) we can say that limits (TL TH) from this sample represent the 95% confidence interval within which 95% of the all possible temperatures lie.

**Example 1:**

A manufacturer of steel bars wants to order boxes for shipping their bars.

They want to order appropriate length for the boxes, with 90% confident that at least 95% of the bars do not exceed the box’s length.

How many samples, N, the manufacturer should select and which one should be used as the measure of the box length?

**Solution:**

From Table I, with γ = 95% and β = 90%, the value for N is 29.

The manufacturer should orders box’s length as the x29 sampled bar (when samples are ordered).

To compare Wilks tolerance limit with Bayes’ Monte Carlo consider a complex Mathematical-based routine [7] (called MDFracture) used to calculate the probability of a nuclear reactor pressure vessel fracture due to pressurized thermal shock.

Certain transient scenarios can cause a rapid cooling inside the reactor vessel while it is pressurized.

**Example 2:**

A 2.828-inch surge line break in a certain design of nuclear reactors may lead to such a condition.

Many input variables contribute to the amount of thermal stress and fracture toughness of the vessel.

Some of them may involve uncertainties.

The temperature, pressure and heat transfer coefficient are examples of such variables, represented by normal distributions.

Also, flaw size, the distance from the flaw inner tip to the interface between base and clad of reactor vessel (C_Dist)) and aspect ratio are unknown and can be represented by random variables with the distributions shown in the Table III.

To compare the results of vessel fracture due to this scenario using Wilks approach with γ = 95% and B = 95% with the results of the standard 1000 and 2000 trials standard Monte Carlo simulation, three Wilks’ runs with 100 samples (assuming γ = 95% and β = 95% with two-sided case as shown in Table II) and two Monte Carlo runs with 1000 and 2000 are performed using the MD-Fracture Mathematical-based tool.

Results show good agreement between Wilks tolerance limits and simple Monte Carlo sampling, as shown in Figure I

**References:**

1) Wilks, S.S., Determination of Sample Sizes for Setting Tolerance Limits. The Annals of Mathematical Statistics, 12(1), 91, 1941.

2) Wilks, S.S., Statistical Prediction with Special Reference to the Problem of Tolerance Limits. The Annals of Mathematical Statistics, 13(4), 400, 1942.

3) Wald, A., An Extension of Wilks’ Method for Setting Tolerance Limits. The Annals of Mathematical Statistics, 14(1), 45, 1943.

4) Wald, A., Tolerance Limits for a Normal Distribution. The Annals of Mathematical Statistics, 17(2), 208, 1946.

5) Guba, A., Makai, M., and Pal, L., Statistical aspects of best estimate method I., Reliability Engineering & System Safety, 80 (3), 217, 2003.

6) Nutt, W.T., and Wallis, G.B., Evaluation of nuclear safety from the outputs of computer codes in the presence of uncertainties. Reliability Engineering & System Safety, 83(1), 57, 2004.

7) Li, F. and Modarres, M., Characterization of Uncertainty in the Measurement of Nuclear Reactor Vessel Fracture Toughness and Probability of Vessel Failure, Transactions of the American Nuclear Society Annual Meeting, Milwaukee, 2001.

By: Mohammad Pourgol-Mohammad, Ph.D, P.E, CRE, mpourgol@gmail.com

Previously published in the December June 2016 Volume 7, Issue 2 ASQ Reliability Division Newsletter

Picture © B. Poncelet https://bennyponcelet.wordpress.com